Question

Evaluate , `lim_(xto1//2) ((8x-3)/(2x-1)-(4x^(2)+1)/(4x^(2)-1))`

Answer 1

Given, `underset(xto1//2)"lim"((8x-3)/(2x-1)-(4x^(2)+1)/(4x^(2)-1))= underset(xto1//2)"lim"[[(8x-3)(2x+1)-(4x^(2)+1))/(4x^(2)-1)]`
`=underset(xto1//2)"lim"[(16x^(2)+8x-6x-3-4x^(2)-1)/(4x^(2)-1)]`
`=underset(xto1//2)"lim"[(12x^(2)+2x-4)/(4x^(2)-1)]`
`=underset(xto1//2)"lim"(2(6x^(2)+x-2))/(4x^(2)-1)`
`=undersett(xto1//2)"lim"(2(6x^(2)+4x-3x-2))/(4x^(2)-1)`
`=underset(xto1//2)"lim"(2[2x(3x+2)-1(3x+2))]/(4x^(2)-1)`
`=underset(xto1//2)"lim"(2(3x+2)(2x-1))/((2x-1)(2x+1))`
`=underset(xto1//2)"lim"(2(3x+2))/(2x-1) = (2(3 xx 1/2 +2))/(2 xx 1/2+1)`
`=3/2+2=7/2`